LMFDB - Embedded newform 2400.2.d.d.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,2,Mod(49,2400)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2400.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$19.1640964851$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2400.49
Dual form			2400.2.d.d.49.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.00000i q^{7} +1.00000 q^{9} -4.00000i q^{11} +6.00000i q^{17} +4.00000i q^{19} +2.00000i q^{21} +4.00000i q^{23} +1.00000 q^{27} +6.00000i q^{29} -10.0000 q^{31} -4.00000i q^{33} +4.00000 q^{37} +10.0000 q^{41} -4.00000 q^{43} -4.00000i q^{47} +3.00000 q^{49} +6.00000i q^{51} +10.0000 q^{53} +4.00000i q^{57} +8.00000i q^{59} -8.00000i q^{61} +2.00000i q^{63} +12.0000 q^{67} +4.00000i q^{69} +4.00000 q^{71} +10.0000i q^{73} +8.00000 q^{77} -14.0000 q^{79} +1.00000 q^{81} +6.00000i q^{87} -14.0000 q^{89} -10.0000 q^{93} +10.0000i q^{97} -4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{9} + 2 q^{27} - 20 q^{31} + 8 q^{37} + 20 q^{41} - 8 q^{43} + 6 q^{49} + 20 q^{53} + 24 q^{67} + 8 q^{71} + 16 q^{77} - 28 q^{79} + 2 q^{81} - 28 q^{89} - 20 q^{93}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^9 + 2 * q^27 - 20 * q^31 + 8 * q^37 + 20 * q^41 - 8 * q^43 + 6 * q^49 + 20 * q^53 + 24 * q^67 + 8 * q^71 + 16 * q^77 - 28 * q^79 + 2 * q^81 - 28 * q^89 - 20 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$-1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	− 4.00000i	− 1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$11$	− 4.00000i	− 1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	0	0
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	0	0
$21$	2.00000i	0.436436i
$22$	0	0
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.00000i	1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$	6.00000i	1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	0	0
$33$	− 4.00000i	− 0.696311i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 4.00000i	− 0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$	− 4.00000i	− 0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	6.00000i	0.840168i
$52$	0	0
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.00000i	0.529813i
$58$	0	0
$59$	8.00000i	1.04151i	0.853706	+	0.520756i	$0.174350\pi$
$59$	8.00000i	1.04151i	−0.853706	+	0.520756i	$0.825650\pi$
$60$	0	0
$61$	− 8.00000i	− 1.02430i	−0.858898	−	0.512148i	$-0.828850\pi$
$61$	− 8.00000i	− 1.02430i	0.858898	−	0.512148i	$-0.171150\pi$
$62$	0	0
$63$	2.00000i	0.251976i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	4.00000i	0.481543i
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.00000	0.911685
$78$	0	0
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.00000i	0.643268i
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−10.0000	−1.03695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	0	0
$99$	− 4.00000i	− 0.402015i
$100$	0	0
$101$	14.0000i	1.39305i	0.717532	+	0.696526i	$0.245272\pi$
$101$	14.0000i	1.39305i	−0.717532	+	0.696526i	$0.754728\pi$
$102$	0	0
$103$	2.00000i	0.197066i	0.995134	+	0.0985329i	$0.0314150\pi$
$103$	2.00000i	0.197066i	−0.995134	+	0.0985329i	$0.968585\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	− 4.00000i	− 0.383131i	−0.981480	−	0.191565i	$-0.938644\pi$
$109$	− 4.00000i	− 0.383131i	0.981480	−	0.191565i	$-0.0613564\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−12.0000	−1.10004
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	10.0000	0.901670
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 6.00000i	− 0.532414i	−0.963916	−	0.266207i	$-0.914230\pi$
$127$	− 6.00000i	− 0.532414i	0.963916	−	0.266207i	$-0.0857705\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.00000i	0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$	2.00000i	0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$	0	0
$139$	− 4.00000i	− 0.339276i	−0.985506	−	0.169638i	$-0.945740\pi$
$139$	− 4.00000i	− 0.339276i	0.985506	−	0.169638i	$-0.0542598\pi$
$140$	0	0
$141$	− 4.00000i	− 0.336861i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.00000	0.247436
$148$	0	0
$149$	6.00000i	0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$	6.00000i	0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	6.00000i	0.485071i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	20.0000	1.59617	0.798087	−	0.602542i	$-0.205846\pi$
$157$	20.0000	1.59617	0.798087	+	0.602542i	$0.205846\pi$
$158$	0	0
$159$	10.0000	0.793052
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	24.0000i	1.85718i	0.371113	+	0.928588i	$0.378976\pi$
$167$	24.0000i	1.85718i	−0.371113	+	0.928588i	$0.621024\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	4.00000i	0.305888i
$172$	0	0
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8.00000i	0.601317i
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	− 20.0000i	− 1.48659i	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	− 20.0000i	− 1.48659i	0.668965	−	0.743294i	$-0.266738\pi$
$182$	0	0
$183$	− 8.00000i	− 0.591377i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	24.0000	1.75505
$188$	0	0
$189$	2.00000i	0.145479i
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	0	0
$199$	−6.00000	−0.425329	−0.212664	−	0.977125i	$-0.568214\pi$
$199$	−6.00000	−0.425329	−0.212664	+	0.977125i	$0.568214\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	0	0
$203$	−12.0000	−0.842235
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.00000i	0.278019i
$208$	0	0
$209$	16.0000	1.10674
$210$	0	0
$211$	− 12.0000i	− 0.826114i	−0.910705	−	0.413057i	$-0.864461\pi$
$211$	− 12.0000i	− 0.826114i	0.910705	−	0.413057i	$-0.135539\pi$
$212$	0	0
$213$	4.00000	0.274075
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 20.0000i	− 1.35769i
$218$	0	0
$219$	10.0000i	0.675737i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 10.0000i	− 0.669650i	−0.942280	−	0.334825i	$-0.891323\pi$
$223$	− 10.0000i	− 0.669650i	0.942280	−	0.334825i	$-0.108677\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	4.00000i	0.264327i	0.991228	+	0.132164i	$0.0421925\pi$
$229$	4.00000i	0.264327i	−0.991228	+	0.132164i	$0.957808\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−14.0000	−0.909398
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	−30.0000	−1.93247	−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−30.0000	−1.93247	−0.966235	+	0.257663i	$0.917048\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 12.0000i	− 0.757433i	−0.925513	−	0.378717i	$-0.876365\pi$
$251$	− 12.0000i	− 0.757433i	0.925513	−	0.378717i	$-0.123635\pi$
$252$	0	0
$253$	16.0000	1.00591
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.00000i	0.124757i	0.998053	+	0.0623783i	$0.0198685\pi$
$257$	2.00000i	0.124757i	−0.998053	+	0.0623783i	$0.980131\pi$
$258$	0	0
$259$	8.00000i	0.497096i
$260$	0	0
$261$	6.00000i	0.371391i
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−14.0000	−0.856786
$268$	0	0
$269$	− 10.0000i	− 0.609711i	−0.952399	−	0.304855i	$-0.901392\pi$
$269$	− 10.0000i	− 0.609711i	0.952399	−	0.304855i	$-0.0986081\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−16.0000	−0.961347	−0.480673	−	0.876900i	$-0.659608\pi$
$277$	−16.0000	−0.961347	−0.480673	+	0.876900i	$0.659608\pi$
$278$	0	0
$279$	−10.0000	−0.598684
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	20.0000i	1.18056i
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	10.0000i	0.586210i
$292$	0	0
$293$	2.00000	0.116841	0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000	0.116841	0.0584206	+	0.998292i	$0.481394\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 4.00000i	− 0.232104i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 8.00000i	− 0.461112i
$302$	0	0
$303$	14.0000i	0.804279i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	0	0
$309$	2.00000i	0.113776i
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	6.00000i	0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$	6.00000i	0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	24.0000	1.34374
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	−24.0000	−1.33540
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 4.00000i	− 0.221201i
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	− 4.00000i	− 0.219860i	−0.993939	−	0.109930i	$-0.964937\pi$
$331$	− 4.00000i	− 0.219860i	0.993939	−	0.109930i	$-0.0350627\pi$
$332$	0	0
$333$	4.00000	0.219199
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 30.0000i	− 1.63420i	−0.576493	−	0.817102i	$-0.695579\pi$
$337$	− 30.0000i	− 1.63420i	0.576493	−	0.817102i	$-0.304421\pi$
$338$	0	0
$339$	6.00000i	0.325875i
$340$	0	0
$341$	40.0000i	2.16612i
$342$	0	0
$343$	20.0000i	1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.0000	−0.858925	−0.429463	−	0.903085i	$-0.641297\pi$
$347$	−16.0000	−0.858925	−0.429463	+	0.903085i	$0.641297\pi$
$348$	0	0
$349$	− 8.00000i	− 0.428230i	−0.976808	−	0.214115i	$-0.931313\pi$
$349$	− 8.00000i	− 0.428230i	0.976808	−	0.214115i	$-0.0686868\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 18.0000i	− 0.958043i	−0.877803	−	0.479022i	$-0.840992\pi$
$353$	− 18.0000i	− 0.958043i	0.877803	−	0.479022i	$-0.159008\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−12.0000	−0.635107
$358$	0	0
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	−5.00000	−0.262432
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 18.0000i	− 0.939592i	−0.882775	−	0.469796i	$-0.844327\pi$
$367$	− 18.0000i	− 0.939592i	0.882775	−	0.469796i	$-0.155673\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	0	0
$371$	20.0000i	1.03835i
$372$	0	0
$373$	−20.0000	−1.03556	−0.517780	−	0.855514i	$-0.673242\pi$
$373$	−20.0000	−1.03556	−0.517780	+	0.855514i	$0.673242\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 36.0000i	− 1.84920i	−0.380945	−	0.924598i	$-0.624401\pi$
$379$	− 36.0000i	− 1.84920i	0.380945	−	0.924598i	$-0.375599\pi$
$380$	0	0
$381$	− 6.00000i	− 0.307389i
$382$	0	0
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	− 10.0000i	− 0.507020i	−0.967333	−	0.253510i	$-0.918415\pi$
$389$	− 10.0000i	− 0.507020i	0.967333	−	0.253510i	$-0.0815851\pi$
$390$	0	0
$391$	−24.0000	−1.21373
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	20.0000	1.00377	0.501886	−	0.864934i	$-0.332640\pi$
$397$	20.0000	1.00377	0.501886	+	0.864934i	$0.332640\pi$
$398$	0	0
$399$	−8.00000	−0.400501
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 16.0000i	− 0.793091i
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	2.00000i	0.0986527i
$412$	0	0
$413$	−16.0000	−0.787309
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 4.00000i	− 0.195881i
$418$	0	0
$419$	− 4.00000i	− 0.195413i	−0.995215	−	0.0977064i	$-0.968849\pi$
$419$	− 4.00000i	− 0.195413i	0.995215	−	0.0977064i	$-0.0311506\pi$
$420$	0	0
$421$	20.0000i	0.974740i	0.873195	+	0.487370i	$0.162044\pi$
$421$	20.0000i	0.974740i	−0.873195	+	0.487370i	$0.837956\pi$
$422$	0	0
$423$	− 4.00000i	− 0.194487i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	16.0000	0.774294
$428$	0	0
$429$	0	0
$430$	0	0
$431$	36.0000	1.73406	0.867029	−	0.498257i	$-0.166026\pi$
$431$	36.0000	1.73406	0.867029	+	0.498257i	$0.166026\pi$
$432$	0	0
$433$	14.0000i	0.672797i	0.941720	+	0.336399i	$0.109209\pi$
$433$	14.0000i	0.672797i	−0.941720	+	0.336399i	$0.890791\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−16.0000	−0.765384
$438$	0	0
$439$	2.00000	0.0954548	0.0477274	−	0.998860i	$-0.484802\pi$
$439$	2.00000	0.0954548	0.0477274	+	0.998860i	$0.484802\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	16.0000	0.760183	0.380091	−	0.924949i	$-0.375893\pi$
$443$	16.0000	0.760183	0.380091	+	0.924949i	$0.375893\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.00000i	0.283790i
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	− 40.0000i	− 1.88353i
$452$	0	0
$453$	2.00000	0.0939682
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 18.0000i	− 0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$	− 18.0000i	− 0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$	0	0
$459$	6.00000i	0.280056i
$460$	0	0
$461$	− 34.0000i	− 1.58354i	−0.610821	−	0.791769i	$-0.709160\pi$
$461$	− 34.0000i	− 1.58354i	0.610821	−	0.791769i	$-0.290840\pi$
$462$	0	0
$463$	22.0000i	1.02243i	0.859454	+	0.511213i	$0.170804\pi$
$463$	22.0000i	1.02243i	−0.859454	+	0.511213i	$0.829196\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	24.0000i	1.10822i
$470$	0	0
$471$	20.0000	0.921551
$472$	0	0
$473$	16.0000i	0.735681i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	10.0000	0.457869
$478$	0	0
$479$	4.00000	0.182765	0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000	0.182765	0.0913823	+	0.995816i	$0.470871\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−8.00000	−0.364013
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 30.0000i	− 1.35943i	−0.733476	−	0.679715i	$-0.762104\pi$
$487$	− 30.0000i	− 1.35943i	0.733476	−	0.679715i	$-0.237896\pi$
$488$	0	0
$489$	20.0000	0.904431
$490$	0	0
$491$	40.0000i	1.80517i	0.430507	+	0.902587i	$0.358335\pi$
$491$	40.0000i	1.80517i	−0.430507	+	0.902587i	$0.641665\pi$
$492$	0	0
$493$	−36.0000	−1.62136
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.00000i	0.358849i
$498$	0	0
$499$	− 20.0000i	− 0.895323i	−0.894203	−	0.447661i	$-0.852257\pi$
$499$	− 20.0000i	− 0.895323i	0.894203	−	0.447661i	$-0.147743\pi$
$500$	0	0
$501$	24.0000i	1.07224i
$502$	0	0
$503$	12.0000i	0.535054i	0.963550	+	0.267527i	$0.0862064\pi$
$503$	12.0000i	0.535054i	−0.963550	+	0.267527i	$0.913794\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−13.0000	−0.577350
$508$	0	0
$509$	− 18.0000i	− 0.797836i	−0.916987	−	0.398918i	$-0.869386\pi$
$509$	− 18.0000i	− 0.797836i	0.916987	−	0.398918i	$-0.130614\pi$
$510$	0	0
$511$	−20.0000	−0.884748
$512$	0	0
$513$	4.00000i	0.176604i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−16.0000	−0.703679
$518$	0	0
$519$	10.0000	0.438951
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 60.0000i	− 2.61364i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	8.00000i	0.347170i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 12.0000i	− 0.516877i
$540$	0	0
$541$	− 44.0000i	− 1.89171i	−0.324593	−	0.945854i	$-0.605227\pi$
$541$	− 44.0000i	− 1.89171i	0.324593	−	0.945854i	$-0.394773\pi$
$542$	0	0
$543$	− 20.0000i	− 0.858282i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	0	0
$549$	− 8.00000i	− 0.341432i
$550$	0	0
$551$	−24.0000	−1.02243
$552$	0	0
$553$	− 28.0000i	− 1.19068i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−42.0000	−1.77960	−0.889799	−	0.456354i	$-0.849155\pi$
$557$	−42.0000	−1.77960	−0.889799	+	0.456354i	$0.849155\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	24.0000	1.01328
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.00000i	0.0839921i
$568$	0	0
$569$	−2.00000	−0.0838444	−0.0419222	−	0.999121i	$-0.513348\pi$
$569$	−2.00000	−0.0838444	−0.0419222	+	0.999121i	$0.513348\pi$
$570$	0	0
$571$	20.0000i	0.836974i	0.908223	+	0.418487i	$0.137439\pi$
$571$	20.0000i	0.836974i	−0.908223	+	0.418487i	$0.862561\pi$
$572$	0	0
$573$	8.00000	0.334205
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	0	0
$579$	− 14.0000i	− 0.581820i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 40.0000i	− 1.65663i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	− 40.0000i	− 1.64817i
$590$	0	0
$591$	−10.0000	−0.411345
$592$	0	0
$593$	22.0000i	0.903432i	0.892162	+	0.451716i	$0.149188\pi$
$593$	22.0000i	0.903432i	−0.892162	+	0.451716i	$0.850812\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.00000	−0.245564
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	0	0
$603$	12.0000	0.488678
$604$	0	0
$605$	0	0
$606$	0	0
$607$	2.00000i	0.0811775i	0.999176	+	0.0405887i	$0.0129233\pi$
$607$	2.00000i	0.0811775i	−0.999176	+	0.0405887i	$0.987077\pi$
$608$	0	0
$609$	−12.0000	−0.486265
$610$	0	0
$611$	0	0
$612$	0	0
$613$	20.0000	0.807792	0.403896	−	0.914805i	$-0.367656\pi$
$613$	20.0000	0.807792	0.403896	+	0.914805i	$0.367656\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	0	0
$619$	20.0000i	0.803868i	0.915669	+	0.401934i	$0.131662\pi$
$619$	20.0000i	0.803868i	−0.915669	+	0.401934i	$0.868338\pi$
$620$	0	0
$621$	4.00000i	0.160514i
$622$	0	0
$623$	− 28.0000i	− 1.12180i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	16.0000	0.638978
$628$	0	0
$629$	24.0000i	0.956943i
$630$	0	0
$631$	18.0000	0.716569	0.358284	−	0.933613i	$-0.383362\pi$
$631$	18.0000	0.716569	0.358284	+	0.933613i	$0.383362\pi$
$632$	0	0
$633$	− 12.0000i	− 0.476957i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	4.00000	0.158238
$640$	0	0
$641$	14.0000	0.552967	0.276483	−	0.961019i	$-0.410831\pi$
$641$	14.0000	0.552967	0.276483	+	0.961019i	$0.410831\pi$
$642$	0	0
$643$	−12.0000	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.0000i	0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$	12.0000i	0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$	0	0
$649$	32.0000	1.25611
$650$	0	0
$651$	− 20.0000i	− 0.783862i
$652$	0	0
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	10.0000i	0.390137i
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	32.0000i	1.24466i	0.782757	+	0.622328i	$0.213813\pi$
$661$	32.0000i	1.24466i	−0.782757	+	0.622328i	$0.786187\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−24.0000	−0.929284
$668$	0	0
$669$	− 10.0000i	− 0.386622i
$670$	0	0
$671$	−32.0000	−1.23535
$672$	0	0
$673$	− 2.00000i	− 0.0770943i	−0.999257	−	0.0385472i	$-0.987727\pi$
$673$	− 2.00000i	− 0.0770943i	0.999257	−	0.0385472i	$-0.0122730\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−38.0000	−1.46046	−0.730229	−	0.683202i	$-0.760587\pi$
$677$	−38.0000	−1.46046	−0.730229	+	0.683202i	$0.760587\pi$
$678$	0	0
$679$	−20.0000	−0.767530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	48.0000	1.83667	0.918334	−	0.395805i	$-0.129534\pi$
$683$	48.0000	1.83667	0.918334	+	0.395805i	$0.129534\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	4.00000i	0.152610i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 12.0000i	− 0.456502i	−0.973602	−	0.228251i	$-0.926699\pi$
$691$	− 12.0000i	− 0.456502i	0.973602	−	0.228251i	$-0.0733006\pi$
$692$	0	0
$693$	8.00000	0.303895
$694$	0	0
$695$	0	0
$696$	0	0
$697$	60.0000i	2.27266i
$698$	0	0
$699$	− 6.00000i	− 0.226941i
$700$	0	0
$701$	− 22.0000i	− 0.830929i	−0.909610	−	0.415464i	$-0.863619\pi$
$701$	− 22.0000i	− 0.830929i	0.909610	−	0.415464i	$-0.136381\pi$
$702$	0	0
$703$	16.0000i	0.603451i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−28.0000	−1.05305
$708$	0	0
$709$	− 44.0000i	− 1.65245i	−0.563337	−	0.826227i	$-0.690483\pi$
$709$	− 44.0000i	− 1.65245i	0.563337	−	0.826227i	$-0.309517\pi$
$710$	0	0
$711$	−14.0000	−0.525041
$712$	0	0
$713$	− 40.0000i	− 1.49801i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	16.0000	0.597531
$718$	0	0
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	0	0
$723$	−30.0000	−1.11571
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 14.0000i	− 0.519231i	−0.965712	−	0.259616i	$-0.916404\pi$
$727$	− 14.0000i	− 0.519231i	0.965712	−	0.259616i	$-0.0835959\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	− 24.0000i	− 0.887672i
$732$	0	0
$733$	24.0000	0.886460	0.443230	−	0.896408i	$-0.353832\pi$
$733$	24.0000	0.886460	0.443230	+	0.896408i	$0.353832\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 48.0000i	− 1.76810i
$738$	0	0
$739$	12.0000i	0.441427i	0.975339	+	0.220714i	$0.0708386\pi$
$739$	12.0000i	0.441427i	−0.975339	+	0.220714i	$0.929161\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	48.0000i	1.76095i	0.474093	+	0.880475i	$0.342776\pi$
$743$	48.0000i	1.76095i	−0.474093	+	0.880475i	$0.657224\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 8.00000i	− 0.292314i
$750$	0	0
$751$	−26.0000	−0.948753	−0.474377	−	0.880322i	$-0.657327\pi$
$751$	−26.0000	−0.948753	−0.474377	+	0.880322i	$0.657327\pi$
$752$	0	0
$753$	− 12.0000i	− 0.437304i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−8.00000	−0.290765	−0.145382	−	0.989376i	$-0.546441\pi$
$757$	−8.00000	−0.290765	−0.145382	+	0.989376i	$0.546441\pi$
$758$	0	0
$759$	16.0000	0.580763
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	0	0
$763$	8.00000	0.289619
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	6.00000	0.216366	0.108183	−	0.994131i	$-0.465497\pi$
$769$	6.00000	0.216366	0.108183	+	0.994131i	$0.465497\pi$
$770$	0	0
$771$	2.00000i	0.0720282i
$772$	0	0
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.00000i	0.286998i
$778$	0	0
$779$	40.0000i	1.43315i
$780$	0	0
$781$	− 16.0000i	− 0.572525i
$782$	0	0
$783$	6.00000i	0.214423i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	−14.0000	−0.494666
$802$	0	0
$803$	40.0000	1.41157
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 10.0000i	− 0.352017i
$808$	0	0
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	4.00000i	0.140459i	0.997531	+	0.0702295i	$0.0223732\pi$
$811$	4.00000i	0.140459i	−0.997531	+	0.0702295i	$0.977627\pi$
$812$	0	0
$813$	2.00000	0.0701431
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 16.0000i	− 0.559769i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 10.0000i	− 0.349002i	−0.984657	−	0.174501i	$-0.944169\pi$
$821$	− 10.0000i	− 0.349002i	0.984657	−	0.174501i	$-0.0558313\pi$
$822$	0	0
$823$	− 42.0000i	− 1.46403i	−0.681290	−	0.732014i	$-0.738581\pi$
$823$	− 42.0000i	− 1.46403i	0.681290	−	0.732014i	$-0.261419\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4.00000	0.139094	0.0695468	−	0.997579i	$-0.477845\pi$
$827$	4.00000	0.139094	0.0695468	+	0.997579i	$0.477845\pi$
$828$	0	0
$829$	44.0000i	1.52818i	0.645108	+	0.764092i	$0.276812\pi$
$829$	44.0000i	1.52818i	−0.645108	+	0.764092i	$0.723188\pi$
$830$	0	0
$831$	−16.0000	−0.555034
$832$	0	0
$833$	18.0000i	0.623663i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−10.0000	−0.345651
$838$	0	0
$839$	4.00000	0.138095	0.0690477	−	0.997613i	$-0.478004\pi$
$839$	4.00000	0.138095	0.0690477	+	0.997613i	$0.478004\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	−10.0000	−0.344418
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 10.0000i	− 0.343604i
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	16.0000i	0.548473i
$852$	0	0
$853$	44.0000	1.50653	0.753266	−	0.657716i	$-0.228477\pi$
$853$	44.0000	1.50653	0.753266	+	0.657716i	$0.228477\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 6.00000i	− 0.204956i	−0.994735	−	0.102478i	$-0.967323\pi$
$857$	− 6.00000i	− 0.204956i	0.994735	−	0.102478i	$-0.0326771\pi$
$858$	0	0
$859$	− 44.0000i	− 1.50126i	−0.660722	−	0.750630i	$-0.729750\pi$
$859$	− 44.0000i	− 1.50126i	0.660722	−	0.750630i	$-0.270250\pi$
$860$	0	0
$861$	20.0000i	0.681598i
$862$	0	0
$863$	− 24.0000i	− 0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$	− 24.0000i	− 0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−19.0000	−0.645274
$868$	0	0
$869$	56.0000i	1.89967i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	10.0000i	0.338449i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	44.0000	1.48577	0.742887	−	0.669417i	$-0.233456\pi$
$877$	44.0000	1.48577	0.742887	+	0.669417i	$0.233456\pi$
$878$	0	0
$879$	2.00000	0.0674583
$880$	0	0
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	0	0
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.0000i	0.537227i	0.963248	+	0.268614i	$0.0865655\pi$
$887$	16.0000i	0.537227i	−0.963248	+	0.268614i	$0.913434\pi$
$888$	0	0
$889$	12.0000	0.402467
$890$	0	0
$891$	− 4.00000i	− 0.134005i
$892$	0	0
$893$	16.0000	0.535420
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 60.0000i	− 2.00111i
$900$	0	0
$901$	60.0000i	1.99889i
$902$	0	0
$903$	− 8.00000i	− 0.266223i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	52.0000	1.72663	0.863316	−	0.504664i	$-0.168384\pi$
$907$	52.0000	1.72663	0.863316	+	0.504664i	$0.168384\pi$
$908$	0	0
$909$	14.0000i	0.464351i
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−26.0000	−0.857661	−0.428830	−	0.903385i	$-0.641074\pi$
$919$	−26.0000	−0.857661	−0.428830	+	0.903385i	$0.641074\pi$
$920$	0	0
$921$	−28.0000	−0.922631
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.00000i	0.0656886i
$928$	0	0
$929$	−46.0000	−1.50921	−0.754606	−	0.656179i	$-0.772172\pi$
$929$	−46.0000	−1.50921	−0.754606	+	0.656179i	$0.772172\pi$
$930$	0	0
$931$	12.0000i	0.393284i
$932$	0	0
$933$	−8.00000	−0.261908
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 22.0000i	− 0.718709i	−0.933201	−	0.359354i	$-0.882997\pi$
$937$	− 22.0000i	− 0.718709i	0.933201	−	0.359354i	$-0.117003\pi$
$938$	0	0
$939$	6.00000i	0.195803i
$940$	0	0
$941$	− 26.0000i	− 0.847576i	−0.905761	−	0.423788i	$-0.860700\pi$
$941$	− 26.0000i	− 0.847576i	0.905761	−	0.423788i	$-0.139300\pi$
$942$	0	0
$943$	40.0000i	1.30258i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	− 26.0000i	− 0.842223i	−0.907009	−	0.421111i	$-0.861640\pi$
$953$	− 26.0000i	− 0.842223i	0.907009	−	0.421111i	$-0.138360\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	24.0000	0.775810
$958$	0	0
$959$	−4.00000	−0.129167
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	−4.00000	−0.128898
$964$	0	0
$965$	0	0
$966$	0	0
$967$	46.0000i	1.47926i	0.673014	+	0.739630i	$0.265000\pi$
$967$	46.0000i	1.47926i	−0.673014	+	0.739630i	$0.735000\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	− 12.0000i	− 0.385098i	−0.981287	−	0.192549i	$-0.938325\pi$
$971$	− 12.0000i	− 0.385098i	0.981287	−	0.192549i	$-0.0616755\pi$
$972$	0	0
$973$	8.00000	0.256468
$974$	0	0
$975$	0	0
$976$	0	0
$977$	22.0000i	0.703842i	0.936030	+	0.351921i	$0.114471\pi$
$977$	22.0000i	0.703842i	−0.936030	+	0.351921i	$0.885529\pi$
$978$	0	0
$979$	56.0000i	1.78977i
$980$	0	0
$981$	− 4.00000i	− 0.127710i
$982$	0	0
$983$	32.0000i	1.02064i	0.859984	+	0.510321i	$0.170473\pi$
$983$	32.0000i	1.02064i	−0.859984	+	0.510321i	$0.829527\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	8.00000	0.254643
$988$	0	0
$989$	− 16.0000i	− 0.508770i
$990$	0	0
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	0	0
$993$	− 4.00000i	− 0.126936i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−28.0000	−0.886769	−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000	−0.886769	−0.443384	+	0.896332i	$0.646222\pi$
$998$	0	0
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.d.d.49.2		2
3.2	odd	2		7200.2.d.f.2449.2		2
4.3	odd	2		600.2.d.a.349.1		2
5.2	odd	4		480.2.k.a.241.1		2
5.3	odd	4		2400.2.k.b.1201.2		2
5.4	even	2		2400.2.d.a.49.1		2
8.3	odd	2		600.2.d.d.349.1		2
8.5	even	2		2400.2.d.a.49.2		2
12.11	even	2		1800.2.d.h.1549.2		2
15.2	even	4		1440.2.k.a.721.2		2
15.8	even	4		7200.2.k.f.3601.2		2
15.14	odd	2		7200.2.d.e.2449.1		2
20.3	even	4		600.2.k.a.301.2		2
20.7	even	4		120.2.k.a.61.1	✓	2
20.19	odd	2		600.2.d.d.349.2		2
24.5	odd	2		7200.2.d.e.2449.2		2
24.11	even	2		1800.2.d.c.1549.2		2
40.3	even	4		600.2.k.a.301.1		2
40.13	odd	4		2400.2.k.b.1201.1		2
40.19	odd	2		600.2.d.a.349.2		2
40.27	even	4		120.2.k.a.61.2	yes	2
40.29	even	2	inner	2400.2.d.d.49.1		2
40.37	odd	4		480.2.k.a.241.2		2
60.23	odd	4		1800.2.k.g.901.1		2
60.47	odd	4		360.2.k.b.181.2		2
60.59	even	2		1800.2.d.c.1549.1		2
80.27	even	4		3840.2.a.w.1.1		1
80.37	odd	4		3840.2.a.m.1.1		1
80.67	even	4		3840.2.a.d.1.1		1
80.77	odd	4		3840.2.a.r.1.1		1
120.29	odd	2		7200.2.d.f.2449.1		2
120.53	even	4		7200.2.k.f.3601.1		2
120.59	even	2		1800.2.d.h.1549.1		2
120.77	even	4		1440.2.k.a.721.1		2
120.83	odd	4		1800.2.k.g.901.2		2
120.107	odd	4		360.2.k.b.181.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.k.a.61.1	✓	2	20.7	even	4
120.2.k.a.61.2	yes	2	40.27	even	4
360.2.k.b.181.1		2	120.107	odd	4
360.2.k.b.181.2		2	60.47	odd	4
480.2.k.a.241.1		2	5.2	odd	4
480.2.k.a.241.2		2	40.37	odd	4
600.2.d.a.349.1		2	4.3	odd	2
600.2.d.a.349.2		2	40.19	odd	2
600.2.d.d.349.1		2	8.3	odd	2
600.2.d.d.349.2		2	20.19	odd	2
600.2.k.a.301.1		2	40.3	even	4
600.2.k.a.301.2		2	20.3	even	4
1440.2.k.a.721.1		2	120.77	even	4
1440.2.k.a.721.2		2	15.2	even	4
1800.2.d.c.1549.1		2	60.59	even	2
1800.2.d.c.1549.2		2	24.11	even	2
1800.2.d.h.1549.1		2	120.59	even	2
1800.2.d.h.1549.2		2	12.11	even	2
1800.2.k.g.901.1		2	60.23	odd	4
1800.2.k.g.901.2		2	120.83	odd	4
2400.2.d.a.49.1		2	5.4	even	2
2400.2.d.a.49.2		2	8.5	even	2
2400.2.d.d.49.1		2	40.29	even	2	inner
2400.2.d.d.49.2		2	1.1	even	1	trivial
2400.2.k.b.1201.1		2	40.13	odd	4
2400.2.k.b.1201.2		2	5.3	odd	4
3840.2.a.d.1.1		1	80.67	even	4
3840.2.a.m.1.1		1	80.37	odd	4
3840.2.a.r.1.1		1	80.77	odd	4
3840.2.a.w.1.1		1	80.27	even	4
7200.2.d.e.2449.1		2	15.14	odd	2
7200.2.d.e.2449.2		2	24.5	odd	2
7200.2.d.f.2449.1		2	120.29	odd	2
7200.2.d.f.2449.2		2	3.2	odd	2
7200.2.k.f.3601.1		2	120.53	even	4
7200.2.k.f.3601.2		2	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2400.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.00000i q^{7} +1.00000 q^{9} -4.00000i q^{11} +6.00000i q^{17} +4.00000i q^{19} +2.00000i q^{21} +4.00000i q^{23} +1.00000 q^{27} +6.00000i q^{29} -10.0000 q^{31} -4.00000i q^{33} +4.00000 q^{37} +10.0000 q^{41} -4.00000 q^{43} -4.00000i q^{47} +3.00000 q^{49} +6.00000i q^{51} +10.0000 q^{53} +4.00000i q^{57} +8.00000i q^{59} -8.00000i q^{61} +2.00000i q^{63} +12.0000 q^{67} +4.00000i q^{69} +4.00000 q^{71} +10.0000i q^{73} +8.00000 q^{77} -14.0000 q^{79} +1.00000 q^{81} +6.00000i q^{87} -14.0000 q^{89} -10.0000 q^{93} +10.0000i q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{9} + 2 q^{27} - 20 q^{31} + 8 q^{37} + 20 q^{41} - 8 q^{43} + 6 q^{49} + 20 q^{53} + 24 q^{67} + 8 q^{71} + 16 q^{77} - 28 q^{79} + 2 q^{81} - 28 q^{89} - 20 q^{93}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^9 + 2 * q^27 - 20 * q^31 + 8 * q^37 + 20 * q^41 - 8 * q^43 + 6 * q^49 + 20 * q^53 + 24 * q^67 + 8 * q^71 + 16 * q^77 - 28 * q^79 + 2 * q^81 - 28 * q^89 - 20 * q^93

Introduction

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